Abstract: In practice the Galerkin method for solving elliptic partial differential equations yields equations involving certain integrals which cannot be evaluated analytically. Instead these integrals are approximated numerically and the resulting equations are solved to give ``quadrature-Galerkin approximations'' to the solution of the differential equation. Using a technique of J. Nitsche, a priori error bounds are obtained for the difference between the solution of the differential equation and a class of quadrature-Galerkin approximations.